Title: 6.088 Digital and Computational Photography 6.882 Advanced Computational Photography Gradient image processing
16.088 Digital and Computational Photography
6.882 Advanced Computational PhotographyGradi
ent image processing
WarningFrench Mathematicians inside
Frédo DurandMIT - EECS
2How was pset 1?
3What have we learnt last time?
- Log is good
- Luminance is different from chrominance
- Separate components
- Low and high frequencies
- Strong edges are important
4Homomorphic filtering
- Oppenhein, in the sixties
- Images are the product of illumination and albedo
- Similarly, many sounds are the product of an
envelope and a modulation - Illumination is usually slow-varying
- Perform albedo-illumination using low-pass
filtering of the log image - http//www.cs.sfu.ca/stella/papers/blairthesis/ma
in/node33.html - See also Koenderink "Image processing done
right"http//www.springerlink.com/(l1bpumaapconcb
jngteojwqv)/app/home/contribution.asp?referrerpar
entbacktoissue,11,53journal,1538,3333linkingpu
blicationresults,1105633,1
5What's great about the bilateral filter
- Separate image into two components
- Preserve strong edges
- Non-iterative
- More controllable, stable
- Can be accelerated
- Lots of other applications
6Bilateral filtering on meshes
- http//www.cs.tau.ac.il/dcor/online_papers/papers
/shachar03.pdf - http//people.csail.mit.edu/thouis/JDD03.pdf
7Questions?
8Today Gradient manipulation
- Idea
- Human visual system is very sensitive to gradient
- Gradient encode edges and local contrast quite
well - Do your editing in the gradient domain
- Reconstruct image from gradient
- Various instances of this idea, Ill mostly
follow Perez et al. Siggraph 2003 - http//research.microsoft.com/vision/cambridge/p
apers/perez_siggraph03.pdf
r
9Problems with direct cloning
From Perez et al. 2003
10Solution clone gradient
11Gradients and grayscale images
- Grayscale image n n scalars
- Gradient
- Overcomplete!
- Whats up with this?
- Not all vector fields are the gradient of an
image! - Only if they are curl-free (a.k.a. conservative)
- But it does not matter for us
n n 2D vectors
12Today message I
- Manipulating the gradient is powerful
13Today message II
- Optimization is powerful
- In particular least square
- Good Least square optimization reduces to a big
linear system - We are going to spend our time going back and
force between minimization and setting
derivatives to zero. - Your head will spin.
- Linear algebra is your friend
- Big sparse linear systems can be solved
efficiently
14Today message III
- Toy examples are good to further understanding
- 1D can however be overly simplifying, n-D is much
more complicated
15Questions?
16Seamless Poisson cloning
- Given vector field v (pasted gradient), find the
value of f in unknown region that optimize
Poisson equationwith Dirichlet conditions
Pasted gradient
Mask
unknownregion
Background
17Discrete 1D example minimization
- Copy to
- Min ((f2-f1)-1)2
- Min ((f3-f2)-(-1))2
- Min ((f4-f3)-2)2
- Min ((f5-f4)-(-1))2
- Min ((f6-f5)-(-1))2
With f16f61
181D example minimization
- Copy to
- Min ((f2-6)-1)2 gt f2249-14f2
- Min ((f3-f2)-(-1))2 gt f32f221-2f3f2 2f3-2f2
- Min ((f4-f3)-2)2 gt f42f324-2f3f4 -4f44f3
- Min ((f5-f4)-(-1))2 gt f52f421-2f5f4 2f5-2f4
- Min ((1-f5)-(-1))2 gt f524-4f5
191D example big quadratic
- Copy to
- Min (f2249-14f2
- f32f221-2f3f2 2f3-2f2
- f42f324-2f3f4 -4f44f3
- f52f421-2f5f4 2f5-2f4
- f524-4f5) Denote it Q
201D example derivatives
Min (f2249-14f2 f32f221-2f3f2 2f3-2f2
f42f324-2f3f4 -4f44f3 f52f421-2f5f4
2f5-2f4 f524-4f5) Denote it Q
211D example set derivatives to zero
gt
221D example
23Questions?
241D example remarks
- Copy to
- Matrix is sparse
- Matrix is symmetric
- Everything is a multiple of 2
- because square and derivative of square
- Matrix is a convolution (kernel -2 4 -2)
- Matrix is independent of gradient field. Only RHS
is - Matrix is a second derivative
25Questions?
26Lets try to further analyze
27Membrane interpolation
- What if v is null?
- Laplace equation (a.k.a. membrane equation )
281D example minimization
- Minimize derivatives to interpolate
- Min (f2-f1)2
- Min (f3-f2)2
- Min (f4-f3)2
- Min (f5-f4)2
- Min (f6-f5)2
With f16f61
291D example derivatives
- Minimize derivatives to interpolate
Min (f2236-12f2 f32f22-2f3f2
f42f32-2f3f4 f52f42-2f5f4 f521-2f5)
Denote it Q
301D example set derivatives to zero
- Minimize derivatives to interpolate
gt
311D example
- Minimize derivatives to interpolate
- Pretty much says that second derivative should
be zero - (-1 2 -1) is a second derivative filter
32Intuition
- In 1D just linear interpolation!
- The min of s f is the slope integrated over the
interval - Locally, if the second derivative was not zero,
this would mean that the first derivative is
varying, which is bad since we want s f to be
minimized - Note that, in 1D by setting f'', we leave two
degrees of freedom. This is exactly what we need
to control the boundary condition at x1 and x2
x1
x2
33In 2D membrane interpolation
x1
x2
34Membrane interpolation
- What if v is null?
- Laplace equation (a.k.a. membrane equation )
- Mathematicians will tell you there is an
Associated Euler-Lagrange equation - Where the Laplacian ? is similar to -1 2 -1in 1D
- Kind of the idea that we want a minimum, so we
kind of derive and get a simpler equation
35Questions?
36What is v is not null?
37What if v is not null?
Seamlessly paste
onto
Just add a linear function so that the boundary
condition is respected
38(Review) Seamless Poisson cloning
- Given vector field v (pasted gradient), find the
value of f in unknown region that optimize
Poisson equationwith Dirichlet conditions
Pasted gradient
Mask
unknownregion
Background
39What if v is not null 2D
- Variational minimization (integral of a
functional)with boundary condition - Euler-Lagrange equation
- (Compared to Laplace, we have replaced ? 0 by ?
div)
40In 2D, if v is conservative
- If v is the gradient of an image g
- Correction function so that
- performs membrane interpolation over ?
411D example
Add
Result
Difference
Solve Laplace
42In 2D, if v is NOT conservative
- Also need to project the vector field v to a
conservative field - And do the membrane thing
- Of course, we do not need to worry about it, its
all handled naturally by the least square approach
43Questions?
44Recap
- Find image whose gradient best approximates the
input gradient - least square Minimization
- Discrete case turns into linear equation
- Set derivatives to zero
- Derivatives of quadratic gt linear
- Continuous turns into Euler-Lagrange form
- ? f div v
- When gradient is null, membrane interpolation
- Linear interpolation in 1D
45Fourier perspective
- Gradient in Fourier?
- Multiply coeffs by i ?
- Parseval theorem?
- Integral of square is the same in space
frequencysx f(x)2 dx s? F(?)2 d? - Least square on gradient ?
- Least square in Fourier with weight ?
- Tries to respect high frequencies at the
potential cost of low frequencies
46Fourier interpretation
- Least square on gradient
- Parseval anybody?
- Integral of squared stuff is the same in Fourier
and primal - What is the gradient/derivative in Fourier?
- Multiply coefficients by frequency and i
- Seen in Fourier, Poisson editing does a weighted
least square of the image where low frequencies
have a small weight and high frequencies a big
weight
47Questions?
48Warning
- What follows is not strictly necessary to
implement Poisson image editing - But
- It helps understand the properties of the
equation - It helps to read the literature
- It's cool math
49Calculus
- Simplified version
- Want to minimize g(x) over the space of real
values x - Derive and set g'(x)0
- Now we have a more complex equation we want to
minimize a variational equation over the space of
functions f - It's a complex business to derive wrt functions
- In general, derivatives are well defined only for
functions over 1D domains
50Derivative definition
- 1D derivative
- multidimensional derivative
- For a direction v, directional derivative is
- For functionals ?
- Do something similar, replace vector by function
51Calculus of variation 1D
- We want to minimize with
f(x1)a, f(x2)b - Assume we have a solution f
- Try to define some notion of 1D derivative wrt to
a 1D parameter ? in a given direction of
functional space - For a perturbation function ?(x) that also
respects the boundary condition (i.e.
?(x1)?(x2)0)and scalar ?, the integral s
(f'(x)? ?'(x))2 dx should be bigger than for f
alone
52Calculus of variation 1D
- s (f'(x)? ?'(x))2 dx should be bigger than for
f alone - s f'(x) 2 2 ? ?'(x) f'(x) ?2?'(x)2 dx
- The third term is always positive and is
negligible when ? goes to zero - Derive wrt ? and set to zero
- s 2 ?'(x)f'(x) dx 0
53Calculus of variation 1D
- How do we get rid of ? ? And still include the
knowledge that ?(x1)?(x2)0 - When we have an integral of a product and we are
playing with derivatives, look into integration
by parts - Now how do you remember integration by parts?
- Integrate one, derive the other
- It's about the derivative of a product in an
integral
54Calculus of variation 1D
- Integrate by parts
- We know that ?(x1)?(x2)0
- We get
- Must be true for any ?
- Therefore, f''(x) must be zero everywhere
55Summary
- Variational minimization (integral of a
functional)with boundary condition - Derive Euler-Lagrange equation
- Use perturbation function
- Calculus of variation. Set to zero. Integrate by
parts. - Check out the hidden slides for detail
56Questions?
57Discrete solver Recall 1D
gt
58Discrete Poisson solver
- Two approaches
- Minimize variational problem
- Solve Euler-Lagrange equation
- In practice, variational is best
- In both cases, need to discretize derivatives
- Finite differences over 4 pixel neighbors
- We are going to work using pairs
- Partial derivatives are easy on pairs
- Same for the discretization of v
p
q
59Discrete Poisson solver
- Minimize variational problem
- Rearrange and call Np the neighbors of p
- Big yet sparse linear system
Discretized gradient
Discretized v g(p)-g(q)
Boundary condition
(all pairs that are in ?)
Only for boundary pixels
60Discrete Poisson solver
- Minimize variational problem
- Rearrange and call Np the neighbors of p
- Big yet sparse linear system
Discretized gradient
Discretized v g(p)-g(q)
Boundary condition
(all pairs that are in ?)
Only for boundary pixels
61Result (eye candy)
62Questions?
63Recap
- Find image whose gradient best approximates the
input gradient - least square Minimization
- Discrete case turns into big sparse linear
equation - Set derivatives to zero
- Derivatives of quadratic gt linear
64Solving big matrix systems
- Axb
- You can use Matlabs \
- (Gaussian elimination)
- But not very scalable
65Iterative solvers
- Important ideas
- Do not inverse matrix
- Maintain a vector x that progresses towards the
solution - Updates mostly require to apply the matrix.
- In many cases, it means you do no even need to
store the matrix (e.g. for a convolution matrix
you only need the kernel) - Usually, you dont even wait until convergence
- Big questions in which direction do you walk?
- Yes, very similar to gradient descent
66Solving big matrix systems
- Axb, where A is sparse (many zero entries)
- In Pset 3, we ask you to use conjugate gradient
- http//www.cs.cmu.edu/quake-papers/painless-conju
gate-gradient.pdf - http//www.library.cornell.edu/nr/bookcpdf/c10-6.p
df
67Conjugate gradient
- The Conjugate Gradient Method is the most
prominent iterative method for solving sparse
systems of linear equations. Unfortunately, many
textbook treatments of the topic are written with
neither illustrations nor intuition, and their
victims can be found to this day babbling
senselessly in the corners of dusty libraries.
For this reason, a deep, geometric understanding
of the method has been reserved for the elite
brilliant few who have painstakingly decoded the
mumblings of their forebears. Nevertheless, the
Conjugate Gradient Method is a composite of
simple, elegant ideas that almost anyone can
understand. Of course, a reader as intelligent as
yourself will learn them almost effortlessly.
68Axb
- A is square, symmetric and positive-definite
- When A is dense, youre stuck, use
backsubstitution - When A is sparse, iterative techniques (such as
Conjugate Gradient) are faster and more memory
efficient - Simple example
- (Yeah yeah, its not sparse)
69Turn Axb into a minimization problem
- Minimization is more logical to analyze iteration
(gradient ascent/descent) - Quadratic form
- c can be ignored because we want to minimize
- Intuition
- the solution of a linear system is always the
intersection of n hyperplanes - Take the square distance to them
- A needs to be positive-definite so that we have a
nice parabola
70Gradient of the quadratic form
- Not our image gradient!
- Multidimensional gradient (as many dim as rows in
matrix)
since
And since A is symmetric
Not surprising we turned Axb into the
quadratic minimization(if A is not symmetric,
conjuagte gradient finds solution for
71Steepest descent/ascent
- Pick gradient direction
- Find optimum in this direction
Gradient direction
Gradient direction
Energy along the gradient
72Residual
- At iteration i, we are at a point x(i)
- Residual r(i)b-Ax(i)
- Cool property of quadratic form residual -
gradient
73Behavior of gradient descent
- Zigzag or goes straight depending if were lucky
- Ends up doing multiple steps in the same direction
74Conjugate gradient
- Smarter choice of direction
- Ideally, step directions should be orthogonal to
one another (no redundancy) - But tough to achieve
- Next best thing make them A-orthogonal
(conjugate)That is, orthogonal when transformed
by A
75Conjugate gradient
- For each step
- Take the residual (gradient)
- Make it A-orthogonal to the previous ones
- Find minimum along this direction
- Plus life is good
- In practice, you only need the previous one
- You can show that the new residual r(i1) is
already A-orthogonal to all previous directions
p but p(i)
76Recap
- Poisson image cloning paste gradient, enforce
boundary condition - Variational formulation
- Also Euler-Lagrange formulation
- Discretize variational version, leads to big but
sparse linear system - Conjugate gradient is a smart iterative technique
to solve it
77Questions?
78(No Transcript)
79(No Transcript)
80Manipulate the gradient
- Mix gradients of g f take the max
81(No Transcript)
82(No Transcript)
83(No Transcript)
84Reduce big gradients
- Dynamic range compression
- See Fattal et al. 2002
85Questions?
86Issues with Poisson cloning
- Colors
- Contrast
- The backgrounds in f g should be similar
87Improvement local contrast
- Use the log
- Or use covariant derivatives (next slides)
88Covariant derivatives Photoshop
- Photoshop Healing brush
- Developed independently from Poisson editing by
Todor Georgiev (Adobe)
From Todor Georgiev's slides http//photo.csail.mi
t.edu/posters/todor_slides.pdf
89Seamless Image Stitching in the Gradient Domain
- Anat Levin, Assaf Zomet, Shmuel Peleg, and Yair
Weisshttp//www.cs.huji.ac.il/alevin/papers/eccv
04-blending.pdfhttp//eprints.pascal-network.org/
archive/00001062/01/tips05-blending.pdf - Various strategies (optimal cut, feathering)
90Photomontage
- http//grail.cs.washington.edu/projects/photomonta
ge/photomontage.pdf
91Elder's edge representation
- http//elderlab.yorku.ca/elder/publications/journ
als/ElderPAMI01.pdf
92Gradient tone mapping
- Fattal et al. Siggraph 2002
Slide from Siggraph 2005 by Raskar (Graphs by
Fattal et al.)
93Gradient attenuation
From Fattal et al.
94Fattal et al. Gradient tone mapping
95Gradient tone mapping
- Socolinsky, D. Dynamic Range Constraints in Image
Fusion and Visualization , in Proceedings of
Signal and Image Processing 2000, Las Vegas,
November 2000.
96Gradient tone mapping
- Socolinsky, D. Dynamic Range Constraints in Image
Fusion and Visualization , in Proceedings of
Signal and Image Processing 2000.
97- Socolinsky, D. and Wolff, L.B., A new paradigm
for multispectral image visualization and data
fusion, IEEE Conference on Computer Vision and
Pattern Recognition (CVPR), Fort Collins, June
1999.
98Retinex
- Land, Land and McCann (inventor/founder of
polaroid) - Theory of lightness perception (albedo vs.
illumination) - Strong gradients come from albedo, illumination
is smooth
99Questions?
100Color2gray
- Use Lab gradient to create grayscale images
101Poisson Matting
- Sun et al. Siggraph 2004
- Assume gradient of F B is negligible
- Plus various image-editing tools to refine matte
102Gradient camera?
- Tumblin et al. CVPR 2005 http//www.cfar.umd.edu/
aagrawal/gradcam/gradcam.html
103Poisson-ish mesh editing
- http//portal.acm.org/citation.cfm?id1057432.1057
456 - http//www.cad.zju.edu.cn/home/xudong/Projects/mes
h_editing/main.htm - http//people.csail.mit.edu/sumner/research/deftra
nsfer/
104Questions?
105Alternative to membrane
Data
- Thin plate minimize second derivative
Membrane interpolation
Thin-plate interpolation
106Inpainting
- More elaborate energy functional/PDEs
- http//www-mount.ee.umn.edu/guille/inpainting.htm
107Key references
- Socolinsky, D. Dynamic Range Constraints in Image
Fusion and Visualization 2000.
http//www.equinoxsensors.com/news.html - Elder, Image editing in the contour domain, 2001
http//elderlab.yorku.ca/elder/publications/journ
als/ElderPAMI01.pdf - Fattal et al. 2002Gradient Domain HDR
Compression http//www.cs.huji.ac.il/7Edanix/hdr/
- Poisson Image Editing Perez et al.
http//research.microsoft.com/vision/cambridge/pap
ers/perez_siggraph03.pdf - Covariant Derivatives and Vision, Todor Georgiev
(Adobe Systems) ECCV 2006
108Poisson, Laplace, Lagrange, Fourier, Monge,
Parseval
- Fourier studied under Lagrange, Laplace Monge,
and Legendre Poisson were around - They all raised serious objections about
Fourier's work on Trigomometric series - http//www.ece.umd.edu/taylor/frame2.htm
- http//www.mathphysics.com/pde/history.html
- http//www-groups.dcs.st-and.ac.uk/history/Mathem
aticians/Fourier.html - http//www.memagazine.org/contents/current/webonly
/wex80905.html - http//www.shsu.edu/icc_cmf/bio/fourier.html
- http//en.wikipedia.org/wiki/Simeon_Poisson
- http//en.wikipedia.org/wiki/Pierre-Simon_Laplace
- http//en.wikipedia.org/wiki/Jean_Baptiste_Joseph_
Fourier - http//www-groups.dcs.st-and.ac.uk/history/Mathem
aticians/Parseval.html
109Refs Laplace and Poisson
- http//www.ifm.liu.se/boser/elma/Lect4.pdf
- http//farside.ph.utexas.edu/teaching/329/lectures
/node74.html - http//en.wikipedia.org/wiki/Poisson's_equation
- http//www.colorado.edu/engineering/CAS/courses.d/
AFEM.d/AFEM.Ch03.d/AFEM.Ch03.pdf
110Gradient image editing refs
- http//research.microsoft.com/vision/cambridge/pap
ers/perez_siggraph03.pdf - http//www.cs.huji.ac.il/alevin/papers/eccv04-ble
nding.pdf - http//www.eg.org/EG/DL/WS/COMPAESTH/COMPAESTH05/0
75-081.pdf.abstract.pdf - http//photo.csail.mit.edu/posters/Georgiev_Covari
ant.pdf - Covariant Derivatives and Vision, Todor Georgiev
(Adobe Systems) ECCV 2006 - http//www.mpi-sb.mpg.de/hitoshi/research/image_r
estoration/index.shtml - http//www.cs.tau.ac.il/tommer/vidoegrad/
- http//ieeexplore.ieee.org/search/wrapper.jsp?arnu
mber1467600 - http//grail.cs.washington.edu/projects/photomonta
ge/ - http//www.cfar.umd.edu/aagrawal/iccv05/surface_r
econstruction.html - http//www.merl.com/people/raskar/Flash05/
- http//research.microsoft.com/carrot/new_page_1.h
tm - http//www.idiom.com/zilla/Work/scatteredInterpol
ation.pdf
111Links
- How to Get Your SIGGRAPH Paper Rejected, Jim
Kajiya, SIGGRAPH 1993 Papers Chair, (link) - Ted Adelson's Informal guidelines for writing a
paper, 1991. (link) - Notes on technical writing, Don Knuth, 1989.
(pdf) - What's wrong with these equations, David Mermin,
Physics Today, Oct., 1989. (pdf) - Ten Simple Rules for Mathematical Writing,
Dimitri P. Bertsekas (link) - Advice on Research and Writing (at CMU)
- How (and How Not) to Write a Good Systems Paper
by Roy Levin and David D. Redell - Things I Hope Not to See or Hear at SIGGRAPH by
Jim Blinn - How to have your abstract rejected
112Poisson image editing
- Two aspects
- When the new gradient is conservative Just
membrane interpolation to ensure boundary
condition - Otherwise allows you to work with
non-conservative vector fields and - Why is it good?
- More weight on high frequencies
- Membrane tries to use low frequencies to match
boundaries conditions - Manipulation of the gradient can be cool (e.g.
max of the two gradients) - Manipulate local features (edge/gradient) and
worry about global consistency later - Smart thing to do work in log domain
- Limitations
- Color shift, contrast shift (depends strongly on
the difference between the two respective
backgrounds)
113Other functionals
- I lied, some people have used smarted energy
functions Todor Georgievs initial
implementation of the Photoshop healing brush.